A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

1, 0, 9, 4, 9, 9, 8, 9, 8, 4, 3, 7, 0, 8, 7, 2, 4, 2, 8, 6, 5, 0, 4, 0, 8, 3, 0, 0, 7, 1, 5, 5, 2, 4, 6, 7, 1, 2, 9, 1, 0, 5, 1, 4, 0, 6, 0, 7, 0, 5, 4, 3, 6, 0, 2, 0, 6, 5, 8, 0, 3, 3, 4, 2, 9, 5, 5, 1, 8, 7, 5, 4, 4, 9, 6, 2, 2, 1, 4, 0, 5, 4, 1, 3, 0, 7

Offset: 1

Author

Clark Kimberling, May 05 2020

Keywords

• nonn
• cons

Comments

Let S and C denote the graphs of y = sin x and y = cos x. For each point (u, sin u) on S, let S(u) be the line normal to S at (u, sin u), and let (snc u, cos(snc u)) be the point of intersection of S(u) and C. Let d(u) be the distance from (u,sin u) to (snc u, cos(snc u)). We call d(u) the u-normal distance from S to C and note that in [0,Pi], there is a unique number u' such that d(u') > d(u) for all real numbers u except those of the form u' + k*Pi. We call d(u') the maximal normal distance between sine and cosine, and we call snc the sine-normal-to-cosine function.

For each (v, cos v) on C, let C(u) be the line normal to C at (v, cos v), and let (cns v, sin(cns v)) be the point of intersection of C(u) and S. Let e(v) be the distance from (v, cos v) to (cns v, sin(cns v)). We call d(v) the v-normal distance from C to S and note that there exists a unique number v' that maximizes e, and e(v') = d(u'). We call cns the cosine-normal-to-sine function. The numbers u' and v' are given in A332501 and A332503.

Note that the maximal normal distance (see Example) exceeds the normal distance from (Pi/2,1) in sine to (Pi/2,0) in cosine - possibly a surprise!

Examples

			2.72573705679992524967463858129656... = the number u in [0,2 Pi] such that the line normal to S at (u, sin u) passes through the point (3 Pi/4,0); cf. A332501.
0.4039727532995172093189617400663... = sin u; cf. A086751.
1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.
1.9866519235847646080193264936226... = snc u; cf A332503.
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507, A003957.

Programs

Mathematica

Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,
ImageSize -> 600, PlotLabel -> "sine and cosine"]
t = Table[x = x /. FindRoot[Cos[x] == -x Sec[u] + u Sec[u] + Sin[u], {x, 0}], {u, -2 Pi, 2 Pi, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]
ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc(x))"]
t = Table[x = x /. FindRoot[Sin[x] == x Csc[u] - u Csc[u] + Cos[u], {x, 0.1}], {u, -2 Pi + .01, 2 Pi - .01, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]
ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x))"]
u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120]  (* A332501 *)
y = Sin[u]  (* A086751 *)
d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2]  (* A332500 *)
RealDigits[u][[1]]  (* A332501 *)
RealDigits[y][[1]]  (* A086751 *)
RealDigits[d][[1]]  (* A332500 *)

PARI

my(d=solve(x=0,1,cos(x)-x)); sqrt(d^2+2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Jun 16 2021

Formula

d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).

Equals sqrt(d^2+2-2*sqrt(1-d^2)) where d = A003957. - Gleb Koloskov, Jun 16 2021